How should I design the LSTM architecture for multivariate time series forecasting problems?
I have not come across a model like this yet. BUT

If you have not tried smaller models I'd recommend trying that first.

Justification: this lets you use learning curves to diagnose what to do next. 

Also, you might try starting with a GRU (the LSTM overhead may not be needed).

One Idea for a starting model

Observe that turnout does not affect weather. So the weather forecast model will be independent of the turnout model but not vice versa.

(Note: I'm using RNN to denote which ever recurrent architecture you choose)

Formulation for simultaneous prediction:

Weather

$\text{RNN}_w$ is your weather prediction model

$\hat w_t = \text{RNN}_w(\hat w_{t-1})$ be the predicted weather at time $t$

Turnout: The idea here is that people will or will not go to the beach for various reasons (overcrowded, too hot or stormy etc). So in the beach population prediction task we use all these as features to predict the population at the next time-step. This reduces the problem to any one of the classical models already developed. 

$\text{RNN}_p$ is your turnout (beach population) prediction model

$\hat p_t$ be the predicted turnout at time $t$

$\hat c_t=[\hat p_t, \hat w_t]$ is the concatenation at time t

$\hat p_{t+1} = \text{RNN}_p(\hat c_t)$

Formulation in the presence of weather forecast:

Simply replace $\hat w_t$ with the true weather forecasts $\hat f_t$.

A final warning

Unless you think your RNN is better at weather prediction than the forecasting systems I would not recommend using $\text{RNN}_w$ in a production application.

I hope this helps.